Question

A water jug with a capacity of 20 gallons is 20 percent full. At the end of every third day, water is added to the jug. If the amount of water added is equal to 50 percent of the water in the jug at the beginning of that day, how many days does it take for the jug to be at least \(85 \%\) full? $$\begin{array}{l} a. 4 \\ b. 6 \\ c. 12 \\ d. 15 \\ e. 20 \end{array}$$

Short answer

The water jug would be at least 85 percent full after 12 days.

Step-by-step solution

Understand the Initial Condition

Initially, the jug is 20 percent full which is \(20/100 * 20 = 4\) gallons. Note that the increase happens every 3rd day.

Establish the Increment Rule

Every 3rd day, the amount of water in the jug increases by 50 percent. This implies that it gets multiplied by \(1 + 50/100 = 1.5\). So, the jug's content becomes \(1.5 *\) the jug's content at the beginning of that day.

Calculate the Days for the Jug to be at Least 85% Full

This step entails iterative computation for every third day, until the water level reaches or exceeds 85 percent of the jug’s capacity, which is \(85/100 * 20 = 17\) gallons.

Day 3 Calculation

On the third day, the jug’s contents will be \(4 * 1.5 = 6\) gallons.

Day 6 Calculation

On the sixth day, the jug’s contents will be \(6 * 1.5 = 9\) gallons.

Day 9 Calculation

On the ninth day, the jug’s contents will be \(9 * 1.5 = 13.5\) gallons.

Day 12 Calculation

On the twelfth day, the jug’s contents will be \(13.5 * 1.5 = 20.25\) gallons, which is more than 17 gallons. Therefore, the jug will be at least 85 percent full on the 12th day.

Key concepts

Percentage Calculations

Understanding percentage calculations can make a variety of math problems easier to tackle. In this problem, we start by finding out what percentage of a larger quantity the water level represents. Initially, the water jug is 20% full. This means we calculate 20% of the total capacity of the jug, which is 20 gallons.
To do this calculation, convert the percentage to a decimal \(\frac{20}{100} = 0.2\) and multiply it by the jug's total capacity: \ 0.2 \times 20 = 4 \, \text{gallons}\.
Later, we need to determine what 85% full looks like. We use the same method to find 85% of 20 gallons: \ \frac{85}{100} \times 20 = 17 \, \text{gallons}\.
Percentage calculations help us monitor the jug's target fill level as we move further into the problem.

Algebraic Reasoning

Algebraic reasoning plays a crucial role in solving this water jug problem. Each time we add water, the jug's content increases by 50% of the amount present at the start of that day. So, our task is to find how many cycles (in 3-day intervals) are needed for the water to reach 17 gallons or more.
This is a continuous multiplication scenario where the initial amount increases by a multiplication of 1.5 (which represents a 50% increase). So, we apply the formula for growth \(\text{final value} = \text{initial value} \times rate^n\).

• Initial value: The starting amount, 4 gallons.
• Rate: Each increment is 1.5 times the previous amount.
• n: Represents the number of 3-day intervals.

By applying algebraic reasoning, you can derive how many times you need to apply the increase to reach or exceed the desired capacity of 85% full. Through calculation, this was determined to be on the 12th day.

Capacity and Volume Calculations

When tackling capacity and volume calculations, it's essential to understand the total volume that can be held and the volume required to achieve a specific target. In this problem, the jug has a total capacity of 20 gallons, and we need to find how many additional gallons must be added to reach at least 85% capacity.
On day one, the jug has 4 gallons. To reach at least 85% capacity, or 17 gallons, we need to keep adding water in 50% increments until the jug holds 17 or more gallons.
The frequency and manner of these increments rely on understanding the jug's total volume capacity and the proper calculations of each addition. Calculating these accurately ensures that, at the end of the 12th day, when water has been added at intervals, the jug surpasses the needed 17 gallons.